Harry van Zanten
Harry van Zanten
Vrije Universiteit Amsterdam
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Asymptotic Statistics (2020-2021)

In Asymptotic Statistics we study the asymptotic behaviour of (aspects of) statistical procedures. Here “asymptotic” means that we study limiting behaviour as the number of observations tends to infinity. A first important reason for doing this is that in many cases it is very hard, if not impossible to derive for instance exact distributions of test statistics for fixed sample sizes. Asymptotic results are often easier to obtain. These can then be used to construct tests, or confidence regions that approximately have the correct uncertainty level. Similarly, determining estimators or other procedures that are optimal in a specific sense, for instance in the sense of minimal mean squared error or variance, is often not possible if the number of observations is fixed. Using asymptotic results is it however in many cases possible to exhibit procedures that are asymptotically optimal.

In this course we begin by treating the mathematical machinery from probability theory that is necessary to formulate and prove the statements of asymptotic statistics. Important are the various notions of stochastic convergence and their relations, the law of large numbers and the central limit theorem (which the students are assumed to know), the multivariate normal distribution, and the so-called delta method. We will use these tools to study the asymptotic behaviour of statistical procedures.

It is assumed that students have at least successfully completed introductory courses on probability theory and statistics and courses on linear algebra and multivariate calculus. It is highly recommended to follow a course on measure theoretic probability.

Announcements

Course information

Online teaching

Due to the Covid crisis the entire course will be online this year. Every week the schedule is as follows:

Ivan’s notes about the exercises

Exam

Retake

Old exams

Slides

What we have done so far

LectureDateTopicMaterialExercisesRemarksVideo links
19/9introduction, convergenceslides+notes Sec. 1.1 up to and including Theorem 1.71.1, 1.2, 1.3(i), 1.4, 1.10, 1.15watch hour 1 from minute 13:45hour 1, hour 2
216/9convergencerest Sec. 1.1, Sec. 1.21.11, 1.12, 1.17, 1.28, 1.29, 1.321) Skip proof of Prohorov and Helly.
2) There is an error in 1.28. Try to correct it!
hour 1, hour 2
323/9multivariate normalSec. 2.1-2.42.1, 2.2, 2.5, 2.8, 2.13, 2.16, 2.17hour 1, hour 2
430/9chi square test and delta methodSec. 2.5, 3.12.22, 2.23(i), 3.1, 3.2, 3.3Skip Theorem 2.10.hour 1, hour 2
57/10delta methodSec. 3.2, 3.33.12, 3.18watch hour 1 from 6:25hour 1, hour 2
14/10live Q&A
628/10M-estimators, consistencyChap. 4 up to and including p. 45 + extra material in slides4.1, 4.5, 4.8, 4.11(i), 4.12(i, ii)There is extra material about Glivenko-Cantelli theorems under bracketing in the slideshour 1, hour 2
74/11M-estimators, asymptotic normality, MLErest of Chap. 44.13, 4.14, 4.16, 4.18, 4.25watch hour 1 from 23:55hour 1, hour 2, hour 3
811/11nonparametric estimationChapter 55.1, 5.2, 5.3, 5.6hour 1, hour 2
18/11live Q&AZoom link will be provided
925/11minimax lower bounds 1sections 6.1, 6.26.1Material from the second set of lecture notes, see abovehour 1, hour 2, hour 3
102/12minimax lower bounds 2rest of chapter 66.2-6.7hour 1, hour 2
119/12high-dimensional modelschapter 77.3-7.5, 7.7-7.9last lecturehour 1, hour 2